If ${\displaystyle (X,\Sigma )}$  is a measurable space and ${\displaystyle B}$  is a Banach space over a field ${\displaystyle \mathbb {K} }$  (which is the real numbers ${\displaystyle \mathbb {R} }$  or complex numbers ${\displaystyle \mathbb {C} }$ ), then ${\displaystyle f:X\to B}$  is said to be weakly measurable if, for every continuous linear functional ${\displaystyle g:B\to \mathbb {K} ,}$  the function

A measurable function on a probability space is usually referred to as a random variable (or random vector if it takes values in a vector space such as the Banach space ${\displaystyle B}$ ). Thus, as a special case of the above definition, if ${\displaystyle (\Omega ,{\mathcal {P}})}$  is a probability space, then a function ${\displaystyle Z:\Omega \to B}$  is called a (${\displaystyle B}$ -valued) weak random variable (or weak random vector) if, for every continuous linear functional ${\displaystyle g:B\to \mathbb {K} ,}$  the function

The relationship between measurability and weak measurability is given by the following result, known as Pettis' theorem or Pettis measurability theorem.

A function ${\displaystyle f}$  is said to be almost surely separably valued (or essentially separably valued) if there exists a subset ${\displaystyle N\subseteq X}$  with ${\displaystyle \mu (N)=0}$  such that ${\displaystyle f(X\setminus N)\subseteq B}$  is separable.

In the case that ${\displaystyle B}$  is separable, since any subset of a separable Banach space is itself separable, one can take ${\displaystyle N}$  above to be empty, and it follows that the notions of weak and strong measurability agree when ${\displaystyle B}$  is separable.

• Bochner measurable function
• Bochner integral
• Bochner space – Type of topological space
• Pettis integral
• Vector measure

• Pettis, B. J. (1938). "On integration in vector spaces". Trans. Amer. Math. Soc. 44 (2): 277–304. doi:10.2307/1989973. ISSN 0002-9947. MR 1501970.
• Showalter, Ralph E. (1997). "Theorem III.1.1". Monotone operators in Banach space and nonlinear partial differential equations. Mathematical Surveys and Monographs 49. Providence, RI: American Mathematical Society. p. 103. ISBN 0-8218-0500-2. MR 1422252.